Com simular de forma precisa el moviment del nostre Sistema Solar?

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1 Com simular de forma precisa el moviment del nostre Sistema Solar? A.Farrés J. Laskar M. Gastineau S. Blanes F. Casas J. Makazaga A. Murua Institut de Mécanique Céleste et de Calcul des Éphémérides, Observatoire de Paris Instituto de Matemática Multidisciplinar, Universitat Politècnica de València Institut de Matemàtiques i Aplicacions de Castelló, Universitat Jaume I Konputazio Zientziak eta A.A. saila, Informatika Fakultatea 25 Abril 2013

2 Overview of the Talk 1 Why do we want long-term integrations of the Solar System? 2 The N-Body Problem (Toy model for the Planetary motion) 3 Symplectic Splitting Methods for Hamiltonian Systems 4 Conclusions A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

3 Chaotic Motion of the Solar System Secular equations: 200 Ma: Laskar (1989, 1990) Direct integration: 100 Ma: Sussman and Wisdom (1992) d(t ) d 0 10 T /10 A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

4 A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

5 A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

6 A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

7 Planetary Solution La2004 : numerical, simplified, tuned to DE406 (6000 yr) INPOP : numerical, complete, adjusted to observations. 1 Myr : 6 months of CPU. La2010 : numerical, less simplified, tuned to INPOP (1 Myr ). 250Myr : 18 months of CPU. A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

8 A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

For further information http://www.imcce.

9 For further information A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

10 The Challenge 1 The NUMERICAL PRECISION of the solution. We want to be sure that the precision is not a limiting factor. 2 The SPEED of the algorithm. As La2010a took nearly 18 months to complete 250 Myr. A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

11 The N - Body Problem

12 The N-Body Problem We consider that we have n + 1 particles (n planets + the Sun) interacting between each other due to their mutual gravitational attraction. We consider: u 0, u 1,..., u n and u 0, u 1,..., u n the position and velocities of the n + 1 bodies with respect to the centre of mass. ũ i = m i u i the conjugated momenta. The equations of motion are Hamiltonian: H = 1 2 n i=0 ũ i 2 m i G 0 i<j n m i m j u i u j. (1) Notice that the Hamiltonian is naturally split as H = T (p) + U(q). A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

13 Heliocentric Coordinates We consider relative position of each planet (P i ) with respect to the Sun (P 0 ). } },, r 0 = u 0 r i = u i u 0 r 0 = ũ ũ n r i = ũ i In this set of coordinates the Hamiltonian is naturally split into two part: H H = H Kep + H pert : H H = n i=1 ( 1 2 r i 2 where i,j = r i r j. [ m0 + m i m 0m i ] G m0m ) i + ( ri r j G m ) im j, r i m 0 ij 0<i<j n A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

14 Jacobi Coordinates We consider the position of each planet (P i ) w.r.t. the centre of mass of the previous planets (P 0,..., P i 1 ). v 0 = (m 0u m nu n)/η n v i = u i ( i 1 j=0 m ju j )/η i 1 }, ṽ 0 = ũ ũ n ṽ i = (η i 1 ũ i m i ( i 1 j=0 u j))/η i }. where η i = i j=0 m j. In this set of coordinates the Hamiltonian is naturally split into two part: H J = H Kep + H pert : n ( 1 η i ṽ i 2 H J = G m ) iη i 1 n ( ) + G ηi 1 m i 2 η i 1 m i v i v i m0 m i m j r i ij i=1 where i,j = u i u j. i=2 0<i<j n, A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

15 Jacobi Vs Heliocentric In both cases we have H = H Kep + H pert. But: - H H = H A (p, q) + ε(h B (q) + H C (p)), - H J = H A (p, q) + εh B (q), where H A, H B and H C are integrable on their own. Remarks: the size of the perturbation in Jacobi coordinates is smaller that the size of the perturbation in Heliocentric coordinates, giving a better approximation of the real dynamics. the expressions in Heliocentric coordinates are easier to handle, and do not require a specific order on the planets. A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

16 TEST EXAMPLES We know that the most massive planet (i.e. the one that makes the size of the perturbation grow) is Jupiter. So simple models (2-4 planets) including Jupiter should be considered. We also have a problem with the orbital speed of Mercury. Although it is one of the less massive planets it is by far the fastest one. It has a period of 87.9 days and from our results this reduces enormously the optimal step-size. With this in mind, from now on we will consider the following test examples: 4 planets: Jupiter, Saturn, Uranus and Neptune & Mercury, Venus, Earth and Mars 8 planets: Mercury to Neptune & Venus to Pluto A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

17 Jacobi Vs Heliocentric (size of perturbation) np,case Heliocentric Pert. Jacobi Pert. 2, MV E E-011 2, JS E E-007 4, MM E E-010 4, JN E E-007 8, MN E E-007 8, VP E E-007 9, All E E-007 Table: Size of the perturbation in Heliocentric Vs Jacobi coordinates for different type of planetary configurations. 2planets (Merc. and Venus); 2planets (Jup. and Sat.); 4planets (Merc.-Venus-Earth-Mars); 4planets (Jup.-Sat.-Ura.-Nept.); 8planets (Merc. to Nept.); 8planets (Ven. to Plu.); 9planets (Merc. to Plu.) A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

18 Symplectic Splitting Methods for Hamiltonian Systems

19 Splitting Methods for Hamiltonian Systems Let H(q, p) be a Hamiltonian, where (q, p) are a set of canonical coordinates. dz dt = {H, z} = L Hz, (2) where z = (q, p) and {, } is the Poisson Bracket ({F, G} = F qg p F pg q). The formal solution of Eq. (2) at time t = τ that starts at time t = τ 0 is given by, z(τ) = exp(τl H )z(τ 0). (3) We want to build approximations for exp(τl H ) that preserve the symplectic character. A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

20 Splitting Methods for Hamiltonian Systems The formal solution of Eq. (2) at time t = τ that starts at time t = τ 0 is given by, where A L HA, B L HB. z(τ) = exp(τl H )z(τ 0) = exp[τ(a + B)]z(τ 0). (4) We recall that H A and H B are integrable, hence we can compute exp(τa) and exp(τ B) explicitly. We will construct symplectic integrators, S n (τ), that approximate exp[τ(a + B)] by an appropriate composition of exp(τ A) and exp(τ B): S n (τ) = n exp(a i τa) exp(b i τb) i=1 A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

21 Splitting Methods for Hamiltonian Systems Using the Baker-Campbell-Hausdorff (BHC) formula for the product of two exponential of non-commuting operators X and Y : with exp X exp Y = exp Z, Z = X + Y [X, Y ] ([X, [X, Y ]] [Y, [Y, X ]]) + 1 [X, [Y, [Y, X ]]] +..., 24 and [X, Y ] := XY YX. This ensures us that is we have an nth order integrating scheme: k exp(a i τa) exp(b i τb) = exp(τd H ). i=1 Then H = H + τ n H n + o(τ n ) and the error in energy is of order τ n. A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

22 Two simple examples S 1 (τ) = exp(τa) exp(τb), K = A + B + τ 2 τ [A, B] + ([A, [A, B]] + [B, [B, A]]) S 2 (τ) = exp(τ/2a) exp(τb) exp(τ/2a) (Leap-Frog), K = A + B + τ 2 ([A, [A, B]] + [B, [B, A]]) A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

23 Many Authors like Ruth(1983), Neri (1987) and Yoshida(1990) among others have found appropriate set coefficientscients a i, b i in order to have a High Order symplectic integrator (4th, 6th, 8th,...). From now on we will focus on the special case H = H A + εh B, where H A and H B are integrable on its own. This is the case of the N-body planetary system, where the system can be expressed as a Keplerian motion plus a small perturbation due to their mutual interaction. A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

24 Splitting Methods for Hamiltonian Systems Let us call S n (τ) = exp(τk). Where, n S n (τ) = exp(a i τa) exp(b i τεb) = exp(τk), (5) i=1 The BCH theorem ensures us that K L({A, B}), the Lie algebra generated by A and B, and it can be expanded as a double asymptotic series in τ and ε: τk = τp 1,0A + ετp 1,1B + ετ 2 p 2,1[A, B] + ετ 3 p 3,1[A, [A, B]] + ε 2 τ 3 p 3,2[B, [B, A]] + ετ 4 p 4,1[A, [A, [A, B]]] + ε 2 τ 4 p 4,2[A, [B, [B, A]]] + ε 3 τ 4 p 4,3[B, [B, [B, A]] +..., where p i,j are polynomials in a i and b i. A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

25 Splitting Methods for Hamiltonian Systems If is a splitting method S n (τ) such that K = A + εb + o(τ p ). Then, the coefficients a i, b i must satisfy: p 1,0 = 1, p 1,1 = 1, p i,j = 0, for i = 2,..., p. Remark: It is easy to check that, p 0,1 = a 1 + a a n = 1, p 1,1 = b 1 + b b n = 1. If S n (τ) = S n ( τ) then all the terms of order τ 2k+1 are cancelled out. A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

26 Splitting Methods for Hamiltonian Systems S n(τ) = n exp(a i τa) exp(b i ετb) = exp(τk), i=1 In general ε τ (or at least ε τ), so we are more interested in killing the error terms with small powers of ε. We will find the coefficient a i, b i such that: τk τ(a + εb) = O(ετ s ε 2 τ s ε 3 τ s ε m τ sm+1 ). (6) Definition We will say that the method S n (τ) has n stages if it requires n evaluations of exp(τa) and exp(τb) per step-size. Definition We will say that the method S n (τ) has order (s 1, s 2, s 3,...) if it satisfies Eq. (6). A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

27 Remarks The spliting schemes integrate in an exact way (up to machine accuracy) the approximated Hamiltonian H. We will always integrate using a constant step-size τ such that H = H + O(τ n ). To compare the different methods we will check the variation of energy H(t 0 ) H(t) for different step-sizes τ. We will use as test examples the N-body problem for different planetary configurations. A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

28 SABA n or McLachlan (2n,2) methods McLachlan, 1995; Laskar & Robutel, 2001, considered symmetric schemes that only killed the terms of order τ k ε for k = 1,..., 2n. S m (τ) = exp(a 1 τa) exp(b 1 τb)... exp(b 1 τb) exp(a 1 τa). The main advantages are that: We only need n stages to have a method of order (2n, 2). We can guarantee that for all n the coefficients a i, b i will always be positive. - McLachlan, 1995: Composition methods in the presence of small parameters, BIT 35(2), pp Laskar & Robutel, 2001: High order symplectic integrators for perturbed Hamiltonian systems, Celestial Mechanics and Dynamical Astronomy 80(1), A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

29 SABA n or McLachlan (2n,2) methods McLachlan, 1995; Laskar & Robutel, 2001 id order stages a i b i SABA1 or ABA22 (2, 2) 1 a 1 = 1/2 b 1 = 1 a SABA2 or ABA42 (4, 2) 2 1 = 1/2 3/6 a 2 = b 1 = 1/2 3/3 a SABA3 or ABA62 (6, 2) 3 1 = 1/2 15/10 a 2 = b 1 = 5/18 15/10 b 2 = 4/9 a 1 = 1/ ( 30/70 a 2 = SABA4 or ABA82 (8, 2) ) b 1 = 1/4 30/72 30 /70 b 2 = 1/4 + 30/72 a 3 = /35 Table: Table of coefficients for the ABA, BAB methods of order (2s, 2) for s = 1,..., 4. A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

30 SABA n or McLachlan (2n,2) methods Mer - Ven - Ear - Mar (Jacobi Coord) Jup - Sat - Ura - Nep (Jacobi Coord) -4-6 ABA22 ABA42 ABA62 ABA ABA22 ABA42 ABA62 ABA Figure: Comparison of the performance of the SABA n schemes for Jacobi Coordinates. Using log scale maximum error energy Vs. cost (τ/n). A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

31 SABA n or McLachlan (2n,2) methods As we have seen in the figures above, the main limiting factor of these methods are the terms of order τε 2, which become relevant when τ is small. We recall that in the methods described above we have: K = (A + εb) + ετ 2n p 2n,1[A, [A, [A, B]]] + ε 2 τ 2 p 3,2[B, [B, A]] +..., There are in the literature several options to kill the terms of order τ 2 ε 2 {{A, B}, B}. A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

32 Symplectic Integrator (killing the terms of higher order) Let S 0 (τ) be any of the given symmetric symplectic schemes previously described: S 0(τ) = exp(a 1τA) exp(b 1τB)... exp(b 1τB) exp(a 1τA) = exp(τk), where K = (A + εb) + ετ 2n p 2n,1[A, [A, [A, B]]] + ε 2 τ 2 p 3,2[B, [B, A]] In order to kill the terms of order ε 2 τ 2 we can: 1 Add a corrector term: exp( τ 3 ε 2 c/2l C )S 0 (τ) exp( τ 3 ε 2 c/2l C ). 2 Composition method: S m 0 (τ)s 0(cτ)S m 0 (τ), where c = (2m) 1/3. 3 Add extra stages: S(τ) = m i=1 exp(a iτa) exp(b i τb), with m > n. Hence, the reminder will be τ 2n ε + τ 4 ε 2, having methods of order (2n, 4). A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

33 The corrector term L C This option was proposed by Laskar & Robutel, K = (A + εb) + ε 2 τ 2 p 3,2[B, [B, A]] + ετ 2n p 2n,1[A, [A, [A, B]]] +..., Notice that if A is quadratic in p and B depends only of q then [B, [B, A]] is integrable. We will consider SC n (τ) = exp( τ 3 ε 2 b/2l C )S n (τ) exp( τ 3 ε 2 b/2l C ), with C = {{A, B}, B}. order c ABAn c BABn 1 1/12 1/24 2 (2 3)/24 1/72 3 ( )/648 (13 5 5)/ ( )/64800 REMARK: This procedure only works in Jacobi coordinates. A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

34 Composition method The idea behind this option was first discussed by Yoshida (1990). generalise He showed that if S(τ) is a symplectic methods of order 2k, then it is possible to find a new method of order 2k + 2 by taking where c must satisfy, c 2k = 0. We can generalise these as: where now, c = (2m) 1/(2k+1). S(τ)S(cτ)S(τ), S m (τ)s(cτ)s m (τ), With this simple composition methods we can transform any of the (2s, 2) methods described above to (2s, 4) method. REMARK: This procedure works for both set of coordinates. A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

35 Adding an extra stage (McLachlan (2s,4)) McLachlan discussed the possibility of adding an extra stage to methods of order (2s, 2) in order to get rid of the ε 2 τ 2 terms:. n+1 S(τ) = exp(a i τa) exp(b i τb) i=1 id order stages a i b i ABA64 (6, 4) 4 BAB64 (6, 4) 4 ABA84 (8, 4) 5 BAB84 (8, 4) 5 a 1 = a 2 = a 1 = a 2 = a 3 = a 1 = a 2 = a 3 = b 1 = b 2 = b 3 = b 1 = b 2 = b 3 = b 1 = b 2 = b 3 = Notice that we no longer have positive values for the coefficients a i, b i. A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

36 Jacobi Coordinates (first results) -6-8 Mer - Ven - Ear - Mar (Jacobi Coord) ABA82 S2 m m=2 SC ABA Jup - Sat - Ura - Nep (Jacobi Coord) ABA82 S2 m m=2 SC ABA Figure: Simulations for the inner planets: 4BP case = Merc-Ven-Earth-Mars (left) and Jup-Sat-Ura-Nept (right). Plotting cost vs precision for different integrating schemes. A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

37 Remark 1: Splitting Methods in Heliocentric Coordinates We recall that in Heliocentric coordinates: H(p, q) = H A (p, q) + ε(h B (q) + H C (p)). We can use the same integrating schemes introduced above: n S(τ) = exp(a i τa) exp(b i τ(b + C)), i=1 We can use the approximation: exp(τ(b + C)) = exp(τ/2c) exp(τb) exp(τ/2c). Example (Leap-Frog method): S 1(τ) = exp(τ/2a) exp(τ/2c) exp(τb) exp(τ/2c) exp(τ/2a). REMARK: this introduces an extra error term in the approximation of order ε 3 τ 3. A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

38 Jacobi vs Heliocentric (first results) -6-8 Jup - Sat - Ura - Nep (Jacobi Coord) ABA82 S2 m m=2 SC ABA Jup - Sat - Ura - Nep (Helio Coord) ABA 82 S2 m m=2 ABA A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

39 Heliocentric Coordinates (Improving McLachlan) As we have already discussed, in Heliocentric coordinates, we use exp(τ/2c) exp(τb) exp(τ/2c) to integrate the perturbation part. This introduces in our approximation error terms of order ε 3 τ 2 that can become important for small step-sizes. For instance, the McLachlan methods of order (8, 4) becomes a method of order (8, 4, 2) In order to improve the performance of these scheme, we can add an extra stage to get rid of these term. m+1 exp(a i τa) exp(b i ετb) i=1 We must add the extra condition: b b b 3 m = 0 A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

40 Heliocentric Coordinates (Improving McLachlan) id order n a i b i ABAH84 (8, 4) 5 ABAH844 (8, 4, 4) 6 a 1 = a 2 = a 3 = a 1 = a 2 = a 3 = a 4 = b 1 = b 2 = b 3 = b 1 = b 2 = b 3 = A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

41 Heliocentric Coordinates (Improving McLachlan) Mer - Ven - Ear - Mar (Helio Coord) Jup - Sat - Ura - Nep (Helio Coord) -6 ABA 84 ABAH ABA 84 ABAH A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

42 New Schemes for a generalised stepsize (s 1, s 2,...) In this philosophy, we can always add extra stages in order to kill the desired terms in the error approximation. m S m(τ) = exp(a i τa) exp(b i ετb) i=1 We need: First to decide which are the most relevant terms that might be limiting our splitting scheme. Find the minimal set of coefficients that fulfil our requirements (not trivial). Possible drawbacks: Sometimes to much stages are required and no actual gain in the performance of the scheme is observed. The coefficients a i, b i will no longer be positive. We have no control on their size and this can sometimes produce big rounding error propagation for long term-integration. A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

43 New Schemes for Jacobi Coordinates id order n a i b i a 1 = b ABA82 (8, 2) 4 a 2 = = b a 3 = = ABA84 (8, 4) 5 ABA104 (10, 4) 7 ABA864 (8, 6, 4) 7 ABA1064 (10, 6, 4) 8 a 1 = a 2 = a 3 = a 1 = a 2 = a 3 = a 4 = a 1 = a 2 = a 3 = a 4 = a 1 = a 2 = a 3 = a 4 = a 5 = b 1 = b 2 = b 3 = b 1 = b 2 = b 3 = b 4 = b 1 = b 2 = b 3 = b 4 = b 1 = b 2 = b 3 = b 4 = A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

44 New Schemes for Heliocentric Coordinates id order n a i b i a 1 = b ABAH82 (8, 2) 4 a 2 = = b a 3 = = ABAH84 (8, 4) 5 ABAH844 (8, 4, 4) 6 ABAH864 (8, 6, 4) 8 ABAH1064 (10, 6, 4) 9 a 1 = a 2 = a 3 = a 1 = a 2 = a 3 = a 4 = a 1 = a 2 = a 3 = a 4 = a 5 = a 1 = a 2 = a 3 = a 4 = a 5 = b 1 = b 2 = b 3 = b 1 = b 2 = b 3 = b 1 = b 2 = b 3 = b 4 = b 1 = b 2 = b 3 = b 4 = b 5 = A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

45 Results for Jacobi Mer - Ven - Ear - Mar (Jacobi Coord) Jup - Sat - Ura - Nep (Jacobi Coord) [ short ] -6-8 ABA82 ABA84 ABA104 ABA864 ABA ABA82 ABA84 ABA104 ABA864 ABA A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

46 Results for Heliocentric Mer - Ven - Ear - Mar (Helio Coord) Jup - Sat - Ura - Nep (Helio Coord) -6 ABA 82 ABAH 844 ABAH 864 ABAH ABA 82 ABAH 844 ABAH 864 ABAH A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

47 Results Jacobi Vs Heliocentric (I) Mercury - Venus - Earth - Mars Jupiter - Saturn - Uranus - Neptune -6-8 ABA82 Jb ABA82 He ABA84 Jb ABAH844 He ABA1064 Jb ABAH1064 He -6-8 ABA82 Jb ABA82 He ABA84 Jb ABAH844 He ABA1064 Jb ABAH1064 He A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

48 Results Jacobi Vs Heliocentric (II) Mercury to Neptune -6-8 ABA82 Jb ABA82 He ABA84 Jb ABAH844 He ABA1064 Jb ABAH1064 He A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

49 Final Comments Jacobi coordinates offer better results than Heliocentric coordinates. Adding extra stages in order to improve the error approximation (i.e. methods of order (8, 4, 4), (8, 6, 4), (10, 6, 4),... ) in most of the cases improves the results. The high angular momenta of Mercury is the main limiting factor on the optimal step-size. A. Farrés (IMCCE) Simulant el Sistema Solar Abril 25nd, / 50

50 Thank You for Your Attention

Splitting methods (with processing) for near-integrable problems

Splitting methods (with processing) for near-integrable problems Fernando Casas casas@mat.uji.es www.gicas.uji.es Departament de Matemàtiques Institut de Matemàtiques i Aplicacions de Castelló (IMAC) Universitat